SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED
SOLITONS
DIRK HUNDERTMARK AND YOUNG-RAN LEE
Abstract. This is the second part of a series of papers where we develop rigorous decay estimates for breather solutions of an averaged version of the non-linear
Schrödinger equation. In this part we study the diffraction managed discrete nonlinear Schrödinger equation, an equation which describes coupled waveguide arrays
with periodic diffraction management geometries. We show that, for vanishing
average diffraction, all solutions of the non-linear and non-local diffraction management equation decay super-exponentially. As a byproduct of our method, we
also have a simple proof of existence of diffraction managed solitons in the case of
vanishing average diffraction.
1. Introduction
Solitons, localized coherent structures resulting from a balance of non-linear and
dispersive effects, have been the focus of an intense research activity over the last
decades, see [35, 37]. Besides solitons in the continuum, discrete solitons have emerged
in such diverse areas as solid states physics, some biological systems, Bose-Einstein
condensation, and in discrete non-linear optics, e.g., optical waveguide arrays, [5, 9,
31, 32, 38]. The model describing this range of phenomena is given by the discrete
non–linear Schrödinger equation
i
∂
u(x) + d(ξ)(∆u)(x) + |u(x)|2 u(x) = 0
∂ξ
(1.1)
where for waveguide arrays ξ is the distance along the waveguide, x ∈ Z the location
of the array element, ∆ the discrete Laplacian given by (∆f )(x) = f (x + 1) + f (x −
1) − 2f (x) for all x ∈ Z, and d(ξ) the total diffraction along the waveguide.
Nearly a decade after their theoretical prediction in [7], discrete solitons in an
optical waveguide array were studied experimentally and, as in the continuous case
localized stable non-linear waves where found [10]. Similar to the continuous case,
i.e, non-linear fiber-optics, where the dispersion management technique introduced
by [23] in 1980 turned out to be enormously successful in creating stable low power
pulses by periodically varying the dispersion along the glass–fiber cable, see [1, 14,
15, 18, 21, 22, 28, 26, 29, 40] and the survey article [39], the diffraction management
technique was proposed much more recently in [11] in order to create low power stable
Date: 16 April 2008, version 7-3, file diffms-7-3.tex.
c
2008
by the authors. Faithful reproduction of this article, in its entirety, by any means is
permitted for non-commercial purposes.
1
2
D. HUNDERTMARK AND Y.-R. LEE
discrete pulses by periodically varying the diffraction in discrete optical waveguide
arrays. In this case, the total diffraction d(ξ) along the waveguide is given by
e
d(ξ) = ε−1 d(ξ/ε)
+ dav .
(1.2)
Here dav ≥ 0 is the average component of the diffraction and de its mean zero part.
Note that unlike in the continuum case, the diffraction management technique uses
the geometry of the waveguide to achieve a periodically varying diffraction, see [11].
In the region of strong diffraction management ε is a small positive parameter.
Rescaling t = ξ/ε, (1.1) is equivalent to
∂
e
u + d(t)∆u
+ ε dav ∆u + |u|2 u = 0 .
(1.3)
∂t
For small ε an average equation which describes the slow evolution of solutions of
(1.3) was derived and numerical studies showed that this average equation possesses
stable solutions which evolve nearly periodically when used as initial data in the
diffraction managed non-linear discrete Schrödinger equation, [2, 3, 4]. Normalizing
the period in the fast variable t to one, the average equation for the slow part v of
solutions of (1.3) is given by
i
i
∂
v + εdav ∆v + εQ(v, v, v) = 0
∂t
where
Q(v1 , v2 , v3 ) :=
Z
0
1
Ts−1 Ts v1 Ts v2 Ts v3 ds
(1.4)
(1.5)
Rs
e dξ the solution operator for the free discrete
with Ts := eiD(s)∆ and D(s) = 0 d(ξ)
Schrödinger equation with periodically varying diffraction
∂
e
v = −d(t)∆v.
(1.6)
∂t
One should keep in mind that the variable t denotes the distance along the waveguide.
Physically it makes sense to assume that the diffraction profile de is bounded, or even
piecewise constant along the waveguide. This assumption was made in [27, 30, 34].
For our results we need only to assume that its integral D is bounded over one period
in the fast variable t,
Z t
e dξ < ∞,
τ := sup |D(t)| = sup d(ξ)
(1.7)
i
t∈[0,1]
t∈[0,1]
0
where we normalized, without loss of generality, the period in t to one.
Using the same general method as in the continuum case, see, e.g., [43], the averaged equation (1.4) was derived in [2, 3, 4], where it was expressed in the Fourier
space. The above formulation is from [27, 30]. Note that the non-linear and non-local
equation (1.4) has an associated (averaged) Hamiltonian given by
H(v) := ε
1
dav
hv, −∆vi − Q(v, v, v, v)
2
4
(1.8)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
3
P
with hg, f i := x∈Z g(x)f (x) the usual scalar product on l2 (Z), which, in our convention, is linear in the second component and anti-linear in the first and
Z 1X
Q(v1 , v2 , v3 , v4 ) :=
(Ts v1 )(x)(Ts v2 )(x)(Ts v3 )(x)(Ts v4 )(x) ds.
(1.9)
0 x∈Z
Following the procedure for the continuous case in [43], it was shown in [27] that
over long scales 0 ≤ t ≤ Cε−1 solutions of the non-autonomous equation (1.3) stay
ε-close to solutions of the autonomous average equation (1.4) with the same initial
condition. Thus it is interesting to find stationary solutions of (1.4), which are
precisely the right initial conditions leading to breather-like nearly periodic solutions
of (1.3) on long scales 0 ≤ t ≤ Cε−1 . Making the ansatz v(t, x) = eiεωt f (x) in (1.4)
one arrives at the non-linear and non-local eigenvalue problem
−ωf = −dav ∆f − Q(f, f, f ).
(1.10)
ωf = Q(f, f, f )
(1.11)
Solutions of this equation can be found by minimizing the Hamiltonian H in (1.8)
over functions f ∈ l2 (Z) with a fixed l2 -norm. The problem of constructing such
minimizers for positive average diffraction dav > 0 has been studied in [27, 30] using
a discrete version of Lions’ concentration compactness method [24]. Moreover, using
by now classical arguments, see, [41, 42] or [6], it was noticed in [27, 30] that these
minimizers are so-called orbitally stable, explaining at least in part the strong stability
properties of diffraction management.
Similar as in the continuous case, see [19], proving existence of minimizers for
vanishing average dispersion, dav = 0, i.e., existence of weak solutions f ∈ l2 (Z) of
is much harder and has only recently been established in [34] using Ekeland’s variational principle, [12, 13]. Moreover, it was shown in [34], that the corresponding
minimizer is decaying faster than polynomial, which again yields the orbital stability
of solutions of (1.4) for dav = 0 and initial conditions close to a minimizer.
In this paper we continue our study of regularity properties of the dispersion management technique initiated in [16] and study the decay properties of diffraction
management solitons for vanishing average dispersion, i.e., weak solutions of (1.11).
Our main result is a significant strengthening of the super-polynomial decay result
for diffraction managed solitons in [34].
Theorem 1.1 (Super-exponential decay). Assume that the diffraction profile obeys
(1.7). Then any weak solution f ∈ l2 (Z) of (1.11) decays faster than any exponential.
More precisely, with c = 1 + ln(8τ ),
1
|f (x)| . e− 4 (|x|+1)(ln((|x|+1)/2)−c)
for all x ∈ Z.
Remarks 1.2. (i) In particular, for any 0 < µ < 1/4 Theorem 1.1 yields the bound
|f (x)| . (|x| + 1)−µ(|x|+1)
for all x ∈ Z.
(ii) The bound given in Theorem 1.1 rigorously justifies the theoretical and experimental conclusion of [11], that the diffraction management technique leads to optical
soliton like pulses along a waveguide array which are extremely well-localized along
4
D. HUNDERTMARK AND Y.-R. LEE
the array elements.
(iii) The super-exponential decay given in Theorem 1.1 is in stark contrast to the
continuous case where one has, so far, only super-polynomial bounds on the decay of
dispersion management solitons, see [16]. It is believed that the decay in the continuous case is exponential, see [25] for convincing but non-rigorous arguments.
(iv) Weak solutions of (1.11) are defined as
l2 (Z).
ωhg, f i = hg, Q(f, f, f )i
(1.12)
hf1 , Q(f2 , f3 , f4 )i = Q(f1 , f2 , f3 , f4 )
(1.13)
ωhg, f i = Q(g, f, f, f )
(1.14)
for any g ∈
Recalling the definition (1.9) for the four-linear functional Q, a
short calculation gives
for any fj ∈
l2 (Z),
j = 1, 2, 3, 4. Thus f is a weak solutions of (1.11) if and only if
for all g ∈ l2 (Z).
(v) One easily sees that Q(f, f, f, f ) > 0 as soon as f is not the zero function. Thus
ω = Q(f, f, f, f )/hf, f i > 0 for any non-zero solution of (1.11).
Our second result is a simple proof of existence of weak solutions of (1.11) under
the weak condition (1.7) on the diffraction profile.
Theorem 1.3. Assume that the diffraction profile obeys (1.7). Let λ > 0. There is
an f ∈ l2 (Z) with kf k22 = λ such that
Q(f, f, f, f ) = Pλ := sup Q(g, g, g, g)| g ∈ l2 (Z), kgk22 = λ .
This maximizer f is also a weak solution of the diffraction management equation
(1.11) where ω > 0 is a suitable Lagrange-multiplier.
Remark 1.4. The existence of a maximizer is non-trivial, even for the corresponding
problem with dav > 0, since the equation (1.11), respectively (1.10), is invariant under
translations and so is the corresponding energy functional H given by (1.8). Thus
maximizing sequences for Q, respectively minimizing sequences for H, can very easily
converge to zero weakly. This was overcome using Lions’ concentration compactness
principle in [27, 30] for positive average diffraction and, for vanishing average dispersion, in [34] using Ekeland’s variational principle [12, 13, 17], assuming that the
diffraction profile is piecewise constant. Besides holding under much more general
conditions, our proof is rather direct and, we believe, simple. We show that modulo
translations any maximizing sequence has a strongly convergent subsequence, i.e.,
there is a sequence of shifts such that the shifted sequence, which by the translation
invariance of the problem is also a maximizing sequence, has a strongly convergent
subsequence. Our proof avoids the use of the concentration compactness principle but
relies instead on a discrete version of multi-linear Strichartz estimates, see Corollary
2.9 and Lemma 4.4.
Our paper is organized as follows: In the next section we fix our notation and
develop our basic technical estimates, the discrete versions of bilinear and multi–
linear Strichartz estimates from Lemmata 2.7 and 2.8 and Corollary 2.9. All results
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
5
in Section 2 are valid in arbitrary dimension d ≥ 1. The proof of Theorem 1.1 is
given in Section 3, see Theorem 3.2 and Corollary 3.3. Similar to our study of decay
properties of dispersion managed solitons in [16], the main tool in the proof of our
super-exponential decay Theorem 1.1 is the self-consistency bound from Proposition
3.1 on the tail distribution of weak solutions of the diffraction management equation
(1.11). Our existence proof for diffraction management solitons is given in Section
4. It relies heavily on Lemma 4.4, which follows from the enhanced multi–linear
estimates of Corollary 2.9, and a simple characterization of strong convergence in
l2 (Z), or more generally, strong convergence in lp (Zd ) for 1 ≤ p < ∞, given in
Lemma 4.1.
2. Basic estimates
In this section we consider Zd for arbitrary dimension d ≥ 1. First we introduce
some notation. By N we denote the natural numbers and N0 = N ∪ {0}. Given
n ∈ N0 , we denote n! the factorial, 0! = 1 and (n + 1)! = (n + 1)n!. The integers
are denoted by Z and Zd is the d-fold Euclidian product of Z. lp (Zd ) is the usual
sequence space with norm
1/p
X
kf kp =
|f (x)|p
for 1 ≤ p < ∞
(2.1)
x∈Zd
and
kf k∞ = sup |f (x)|.
(2.2)
x∈Zd
Of course, for p = 2 we get the Hilbert space of square summable sequences indexed
by Zd . In this case we use
X
f (x)g(x)
(2.3)
hf, gi :=
x∈Zd
for the scalar product on l2 (Zd ). Here z is the complex conjugate of a complex number
z. The real and imaginary parts of a complex number are given by Re(z) = 12 (z + z)
1
and Im(z) = 2i
(z − z). Note that in our convention the scalar product given by (2.3)
is linear in the second argument and anti-linear in the first. The discrete Laplacian
on Zd is given by
X
f (x + ν) − 2df (x)
(2.4)
∆f (x) =
Pd
|ν|=1
where we take |x| = j=1 |xj | for the norm on Zd . Since ∆ is a bounded symmetric
operator, eit∆ is the unitary solution operator of the free discrete Schrödinger equation
i∂t u = −∆u
(2.5)
e
i∂t u = −d(t)∆u
(2.6)
on l2 (Zd ); for any f ∈ l2 (Zd ) the function u(t, x) = (eit∆ f )(x) solves (2.5) and
u(0, ·) = f . Note that eit∆ is unitary, in particular, keit∆ f k2 = kf k2 for all f ∈ l2 (Zd ).
Rt
e dξ and Tt := eiD(t)∆ . Thus for any
For a diffraction profile de we set D(t) = 0 d(ξ)
initial condition f ∈ l2 (Zd ) the function u(t, ·) = Tt f solves
6
D. HUNDERTMARK AND Y.-R. LEE
with initial condition u(0, ·) = f . Again, Tt is a unitary operator on l2 (Zd ).
For a function f : Zd → C, its support is given by the set
supp (f ) := {x ∈ Zd : f (x) 6= 0}.
For arbitrary A ⊂ Zd and x ∈ Zd the distance from x to A is given by
and for subsets A, B ⊂
Zd
dist(x, A) := inf(|x − y| : y ∈ A)
their distance is given by
dist(A, B) := inf(dist(x, B) : x ∈ A) = inf(|x − y| : x ∈ A, y ∈ B).
For any operator T : l2 (Zd ) → l2 (Zd ) we denote its kernel by
hx|T |yi := hδx , T δy i
where δy is the Kronecker δ-function
δy (x) :=
1
0
if x = y
.
if x =
6 y
T f (x) =
X
hx|T |yif (y) .
In particular,
y∈Zd
(2.7)
We use the . notation in inequalities, if it is convenient not to specify any constants
in the bounds: for two real-valued functions g, h defined on the same domain, g . h
means that there exists a non-negative constant C such that g(x) ≤ Ch(x) for all x.
The extension of the non-linear and non-local functional Q to l2 (Zd ) is again denoted by Q,
Z 1 X
Q(f1 , f2 , f3 , f4 ) =
(Tt f1 )(x)(Tt f2 )(x)(Tt f3 )(x)(Tt f4 )(x) dt.
(2.8)
0
x∈Zd
The first problem is to show that Q is well-defined on l2 (Zd ). Due to the next lemma
this turns out to be easier than in the continuous case.
Lemma 2.1. Let 1 ≤ p ≤ q ≤ ∞. Then, lp (Zd ) ⊂ lq (Zd ) and
kf kq ≤ kf kp .
Proof. Clearly kf k∞ ≤ kf kp for all f ∈ lp (Zd ) and all 1 ≤ p ≤ ∞.
Let 0 ≤ s ≤ 1. Then for all non-negative sequences (an )n∈Zd ,
X s
X
an ≤
asn .
n∈Zd
This follows from
(a1 + a2 )s =
n∈Zd
a1
a2
a1
a2
+
≤ 1−s + 1−s = as1 + as2
1−s
1−s
(a1 + a2 )
(a1 + a2 )
a1
a2
and induction. Now let 1 ≤ p ≤ q < ∞ and f ∈ lp (Zd ). Then with s = p/q,
X
s
X
p
kf kpp =
|f (n)|qs ≥
|f (n)|q = kf kqs
q = kf kq ,
n∈Zd
n∈Zd
(2.9)
(2.10)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
7
where we used (2.10). This finishes the proof of (2.9).
Corollary 2.2. For any fj ∈ l2 (Zd ), j = 1, . . . , 4, we have
Z 1 X Y
4
4
Y
|Q(f1 , f2 , f3 , f4 )| ≤
|Tt fj (x)| dt ≤
kfj k2
0
x∈Zd j=1
(2.11)
j=1
Proof. Of course, the first inequality is just the triangle inequality. By Hölder’s
inequality followed by Lemma 2.1
4
X Y
x∈Zd
j=1
|Tt fj (x)| ≤
4
Y
j=1
kTt fj k4 ≤
4
Y
j=1
kTt fj k2 =
4
Y
j=1
kfj k2
where we also used that Tt is unitary on l2 (Zd ). Thus (2.11) follows by integrating
this over t on [0, 1].
Remark 2.3. The bound from Corollary 2.2 justifies the ad-hoc formal calculation
for all f, fj ∈ l2 (Zd ) with
hf, Q(f1 , f2 , f3 )i = Q(f, f1 , f2 , f3 )
Q(f1 , f2 , f3 ) :=
Z
0
1
Tt−1 Tt f1 Tt f2 Tt f3 dt.
(2.12)
(2.13)
In particular, this shows that Q(f1 , f2 , f3 ) ∈ l2 (Zd ) whenever fj ∈ l2 (Zd ), and Q
defined in (2.13) is a bounded three linear map from (l2 (Zd ))3 to l2 (Zd ). This is in
contrast to the continuous case, where it is bounded only for d = 1, 2, [16, 33, 43].
Lemma 2.4. For the kernel of the free time evolution eit∆ the bound
sup |hx|eit∆ |yi| ≤ min(1, e4dτ
t∈[−τ,τ ]
(4dτ )|x−y|
)
|x − y|!
(2.14)
holds for all x, y ∈ Zd and 0 ≤ τ < ∞.
P
Proof. The operator ∆ is given by ∆f (x) = |y−x|=1 f (y) − 2df (x) and, using, for
example, the discrete Fourier transform, one sees that 0 ≤ −∆ ≤ 4d and k∆k = 4d.
Now, by symmetry of ∆, eit∆ is unitary, hence one always has |hx|eit∆ |yi| ≤ 1 for any
x, y ∈ Zd and all t. Since ∆ is bounded, the Taylor series for the exponential yields
a strongly converging series for eit∆ . Thus
hx|eit∆ |yi =
∞
X
(it)n
n=0
n!
hx|∆n |yi =
∞
X
n=|x−y|
(it)n
hx|∆n |yi
n!
since hx|∆n |yi =
6 0 if and only if x and y are connected by a path of length at most
n, i.e., |x − y| ≤ n. In particular, using k∆k = 4d,
it∆
|hx|e
|yi| ≤
∞
X
n=|x−y|
∞
X (4d|t|)|x−y|+l
|t|n
k∆kn =
n!
(|x − y| + l)!
l=0
8
D. HUNDERTMARK AND Y.-R. LEE
≤
∞
(4d|t|)|x−y| X (4d|t|)l
(4d|t|)|x−y| 4d|t|
.
=
e
|x − y|!
l!
|x − y|!
l=0
Lemma 2.5. Let s ∈ N. For the free time-evolution eit∆ generated by ∆ the bound
X
(4dτ )2s max(s, 1)d−1
sup |hx|eit∆ |yi|2 .
(2.15)
(s!)2
t∈[−τ,τ ]
y: |x−y|≥s
holds, where the implicit constant depends only on the dimension d and τ .
P
Proof. The number of points y ∈ Zd with |y| = dj=1 |yj | = n can be estimated by
2d nd−1 and with Lemma 2.4 we have
∞
2(n+s)
X
X (4dτ )2|y|
X
8dτ d
d−1 (4dτ )
sup |hx|eit∆ |yi|2 ≤ e8dτ
≤
e
2
(n
+
s)
(|y|!)2
((n + s)!)2
t∈[−τ,τ ]
y: |x−y|≥s
|y|≥s
≤
n=0
∞
d−1 (4dτ )2n
n
(4dτ )2s max(s, 1)d−1 8dτ d X 1
+
e
2
(s!)2
max(s, 1)
(n!)2
n=0
∞
≤
2n
(4dτ )2s sd−1 8dτ d X
d−1 (4dτ )
e
2
(1
+
n)
(s!)2
(n!)2
n=0
where we also used (n + s)! ≥ n!s!. Thus the inequality (2.15) holds with constant
P
2n
d−1 (4dτ )
< ∞.
C = e8dτ 2d ∞
n=0 (1 + n)
(n!)2
Remark 2.6. Estimating the number of points y ∈ Zd with |y| = n by 2d nd−1 is, of
course, a gross over–counting. A tighter estimate can be given as follows: Counting
P
the number of different integers xj ≥ 0 with dj=1 xj = n is equal to distributing
d − 1 separators on n + d − 1 places, i.e,
d
X
n+d−1
d
#{x ∈ N0 :
xj = n} =
d−1
j=1
Since we have 2 choices for the sign, except when some coordinates are zero, we get
the better bound
d
d−1
X
Y
2d
d
d n+d−1
#{y ∈ Z :
|yj | = n} ≤ 2
=
(n + j)
d−1
(d − 1)!
j=1
j=1
Qd−1
2d (1 + ε) d−1
j=1 (1 + j/n) d−1
d
≤
=2
n
n
(d − 1)!
(d − 1)!
for some small ε > 0 when n is large. For our purpose, the rough estimate 2d nd−1 is
good enough.
In the formulation of the next lemma we need some more notation. For any r ∈ R
let
⌈r⌉ := min(z ∈ Z : r ≤ z)
(2.16)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
9
the smallest integer greater than or equal to r.
Lemma 2.7 (Strong Bilinear bound). There exists a constant C depending only on
the dimension d and τ such that for f1 , f2 ∈ l2 (Zd ) and s = dist(supp (f1 ), supp (f2 ))
sup k(eit∆ f1 )(eit∆ f2 )k2 ≤ min(1, C
t∈[−τ,τ ]
max(⌈s/2⌉, 1)d−1 (4dτ )⌈s/2⌉
)kf1 k2 kf2 k2 .
⌈s/2⌉!
(2.17)
Proof. First of all, note that
X
k(eit∆ f1 )(eit∆ f2 )k22 =
|eit∆ f1 (x)|2 |eit∆ f2 (x)|2 ≤ keit∆ f1 k2∞ keit∆ f2 k22 .
x∈Zd
Hence, using Lemma 2.1 and the unicity of eit∆ on l2 (Zd ), we see
k(eit∆ f1 )(eit∆ f2 )k22 ≤ keit∆ f1 k22 keit∆ f2 k22 = kf1 k22 kf2 k22
(2.18)
uniformly in t. Now assume that |t| ≤ τ . Let Ij = supp (fj ), j = 1, 2 and assume,
without loss of generality that s = dist(I1 , I2 ) ≥ 1. Moreover we need the slightly
enlarged sets
I1′ := {x : dist(x, I1 ) ≤ dist(x, I2 ) − 1},
I2′ := {x : dist(x, I1 ) ≥ dist(x, I2 )}.
Note that Ij ⊂ Ij′ , j = 1, 2, and I2′ = Zd \ I1′ . The triangle inequality gives
2dist(x, I2 ) − 1 if x ∈ I1′
s = dist(I1 , I2 ) ≤ dist(x, I1 ) + dist(x, I2 ) ≤
2dist(x, I1 )
if x ∈ I2′
so, since the distance is always an integer, we have
min(dist(I1′ , I2 ), dist(I2′ , I1 )) ≥ ⌈s/2⌉.
(2.19)
Certainly, since I1′ ∪ I2′ = Zd ,
X
X
k(eit∆ f1 )(eit∆ f2 )k22 =
|eit∆ f1 (x)|2 |eit∆ f2 (x)|2 +
|eit∆ f1 (x)|2 |eit∆ f2 (x)|2 .
x∈I1′
x∈I2′
(2.20)
Because of
eit∆ fj (x) =
X
y∈Ij
hx|eit∆ |yifj (y),
the Cauchy–Schwarz inequality implies
|eit∆ fj (x)|2 ≤ kfj k22
X
y∈Ij
|hx|eit∆ |yi|2 .
10
D. HUNDERTMARK AND Y.-R. LEE
Together with (2.19) and the bound from Lemma 2.4, this yields
X
sup
|eit∆ f1 (x)|2 |eit∆ f2 (x)|2
t∈[−τ,τ ] x∈I ′
1
≤ sup
X
t∈[−τ,τ ] x∈I ′
|eit∆ f1 (x)|2 kf2 k22 sup
x∈I1′ y∈I
1
≤ sup
t∈[−τ,τ ]
X
x∈Zd
. kf1 k22 kf2 k22
X
2
|hx|eit∆ |yi|2
X
|eit∆ f1 (x)|2 kf2 k22 sup
x∈Zd y: |x−y|≥⌈s/2⌉
|hx|eit∆ |yi|2
(2.21)
max(⌈s/2⌉, 1)d−1 (4dτ )2⌈s/2⌉
.
(⌈s/2⌉!)2
An identical argument gives
sup
X
t∈[−τ,τ ] x∈I ′
|eit∆ f1 (x)|2 |eit∆ f2 (x)|2 . kf1 k22 kf2 k22
max(⌈s/2⌉, 1)d−1 (4dτ )2⌈s/2⌉
.
(⌈s/2⌉!)2
2
(2.22)
The bounds (2.21) and (2.22) together with (2.20) finish the proof of the Lemma.
Lemma 2.8. For j ∈ {1, 2, 3, 4} let fj ∈ l2 (Zd ). For any choice j, k ∈ {1, 2, 3, 4} let
s = dist(supp (fk ), supp (fl )). Then
sup
t∈[−τ,τ ]
4
4
d−1
⌈s/2⌉ Y
X Y
it∆
(e fj )(x) . max(⌈s/2⌉, 1) (4dτ )
kfj k2
⌈s/2⌉!
d
x∈Z j=1
(2.23)
j=1
where the implicit constant depends only on the dimension d and τ .
Proof. Follows using Cauchy-Schwarz together with Corollary 2.2 and Lemma 2.7.
Corollary 2.9. Let the diffraction profile obey the bound (1.7). Then with c =
1 + ln(8dτ ) we have
|Q(f1 , f2 , f3 , f4 )| . e−s(ln s−c)/2+(d−1) ln(max(s/2,1))
4
Y
j=1
kfj k2
(2.24)
where s = dist(supp (fk ), supp (fl )) for any choice j, k ∈ {1, 2, 3, 4} and the implicit
constant depends only on the dimension d and τ from (1.7).
Remark 2.10. Since for any 0 < δ < 1/2,
e−s(ln s−c)/2+(d−1) ln(max(s/2,1)) . e−δs ln s = s−δs ,
the bound (2.24) implies
|Q(f1 , f2 , f3 , f4 )| . s−δs
4
Y
j=1
kfj k2
for all 0 < δ < 1/2, where the implicit constant depends only on d, δ, and τ .
(2.25)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
11
Proof. Since (1.7) holds, there exists 0 < τ < ∞ such that |D(t)| ≤ τ for all 0 ≤ t ≤ 1.
Thus, since Tt = eiD(t)∆ ,
Z 1 X Y
4
4
X
Y
(Tt fj )(x)
sup
|Q(f1 , f2 , f3 , f4 )| ≤
(Tt fj )(x) dt ≤
0
≤
x∈Zd j=1
X
x∈Zd
x∈Zd
0≤t≤1 j=1
4
Y
it∆
(e fj )(x)
sup
t∈[−τ,τ ] j=1
and the bound from Lemma 2.8 implies
|Q(f1 , f2 , f3 , f4 )| .
4
max(⌈s/2⌉, 1)d−1 (4dτ )⌈s/2⌉ Y
kfj k2
⌈s/2⌉!
.
(2.26)
j=1
An easy proof by induction shows n! ≥
en ln n−n .
Hence, using ⌈s/2⌉ ≥ s/2,
max(⌈s/2⌉, 1)d−1 (4dτ )⌈s/2⌉
. e−s/2 ln(s/2)+s/2 ln(4dτ )+s/2+(d−1) ln(max(s/2,1))
⌈s/2⌉!
= e−s ln(s)/2+cs/2+(d−1) ln(max(s/2,1))
where c = 1 + ln(8dτ ). This proves the bound (2.24).
3. Self-consistency bound and super-exponential decay
As in the continuous case, see [16], the key idea is not to focus on the solution f
directly, but to study its tail distribution defined, for n ∈ N0 , by
X
1/2
|f (x)|2
.
(3.1)
α(n) :=
|x|≥n
The fundamental a-priori estimate for the tail distribution of weak solutions f of
(1.11) is given by the following
Proposition 3.1 (Self-consistency bound). Let f be a weak solution of ωf = Q(f, f, f ).
Then with c = 1 + ln(8τ ), where τ is from the bound (1.7) on the diffraction profile,
α(2n) . α(n)3 + e−(n+1)(ln(n+1)−c)/2 .
(3.2)
In particular, for any 0 < δ < 1/2 the bound
α(2n) . α(n)3 + (n + 1)−δ(n+1)
(3.3)
holds. In (3.2) and (3.3) the implicit constants depend only on ω, δ, kf k2 , and τ .
Proof. Since f is a weak solution of ωf = Q(f, f, f ), we have, by definition,
ωhg, f i = Q(g, f, f, f )
for all g ∈ l2 (Z).
Since
α(2n) =
sup
g∈l2 (Z), kgk2 =1,
supp (g)⊂(−∞,−2n]∪[2n,∞)
|hg, f i|
12
D. HUNDERTMARK AND Y.-R. LEE
we need to estimate Q(g, f, f, f ) for g ∈ l2 (Z) with kgk2 = 1 and supp (g) ⊂
(−∞, −2n] ∪ [2n, ∞). Let In = {−n + 1, . . . n − 1}, Inc its complement and split
f into its low and high parts, f = f< + f> with f< = f χIn and f> = f χInc . Using
the multi-linearity of Q,
Q(g, f, f, f ) = Q(g, f< , f, f ) + Q(g, f> , f, f )
= Q(g, f< , f, f ) + Q(g, f> , f< , f ) + Q(g, f> , f> , f< ) + Q(g, f> , f> , f> ).
(3.4)
The last term is estimated by
|Q(g, f> , f> , f> )| . kgk2 kf> k32 = α(n)3 .
For the first three terms in (3.4) we note that each of them contains one f< . Since
s := dist(supp (g), supp (f< )) is at least n + 1, the enhanced multi-linear estimate
(2.24) from Corollary 2.9 applies and gives, since d = 1, for the first term
|Q(g, f< , f, f )| . e−s(ln s−c)/2 kgk2 kf< k2 kf k22 .
Similar bounds hold for the second and third terms. Collecting terms and using
s ≥ n + 1, we see
α(2n) . ω −1 α(n)3 + e−(n+1)(ln(n+1)−c)/2 α(0)3 + α(0)2 α(n) + α(0)α(n)2
. α(n)3 + e−(n+1)(ln(n+1)−c)/2
since α is a bounded decreasing function. This proves (3.2). Note that the implicit
constant depends only on ω, τ , and α(0) = kf k2 . To prove (3.3) one either argues as
above, but uses (2.25) instead of (2.24), or simply notes that e−(n+1)(ln(n+1)−c)/2 .
(n + 1)−δ(n+1) for any 0 < δ < 1/2.
Theorem 3.2 (Super-exponential decay). Let α be a decreasing non-negative function which obeys the self-consistency bound (3.2) of Proposition 3.1 and decays to
zero at infinity. Then the bound
α(n) . e−
n+1
4
−c)
(ln( n+1
2 )
holds for all n ∈ N0 . Here one can choose c = 1 + ln(8τ ), with τ from the bound (1.7)
on the diffraction profile.
Corollary 3.3 (= Theorem 1.1). For any weak solution of ωf = Q(f, f, f ) and any
0 < µ < 14 the bound
−
holds for all x ∈ Z.
|f (x)| . e
|x|+1
4
|x|+1 ln
2
−c
(3.5)
Proof. Given Theorem 3.2, this follows immediately from |f (x)| ≤ α(|x|).
It remains to prove Theorem 3.2. This is done in two steps. The first is a reduction
of the full super-exponential decay to a slower but still super-exponential decay.
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
13
Lemma 3.4. Let α be a non-negative decreasing function which obeys the selfconsistency bound (3.2). Then the bounds
α(n) . e−
n+1
4
−c)
(ln( n+1
2 )
(3.6)
and
α(n) . (n + 1)−µ0 (n+1)
for some µ0 > 0 are equivalent.
(3.7)
Proof. Of course, the bound (3.6) implies (3.7) for all 0 < µ0 < 1/4. To prove the
converse we will show that if α(n) . (n + 1)−µ(n+1) for some µ > 0 and if 3µ < 1/2
one can boost the decay to
5
α(n) . (n + 1)− 4 µ(n+1) .
(3.8)
Assume this for the moment and assume that (3.7) holds for some µ0 > 0. Let l0 ∈ N0
such that 3(5/4)l0 −1 µ0 < 1/2 ≤ 3(5/4)l0 µ0 . We can iterate (3.8) exactly l0 times to
see
l0
α(n) . (n + 1)−(5/4) µ0 (n+1) .
(3.9)
Plugging the estimate (3.9) into the self–consistency bound (3.2) yields
l0 µ (n+1)
0
α(2n) . (n + 1)−3(5/4)
+ e−(n+1)(ln(n+1)−c)/2 . e−(n+1)(ln(n+1)−c)/2
since 3(5/4)l0 µ0 ≥ 1/2, by assumption. Thus for even n we have the bound
n
n
α(n) . e−( 2 +1)(ln( 2 +1)−c)/2 = e−
n+2
(ln( n+2
)−c)
4
2
(3.10)
and, by monotonicity of α, for odd n the bound (3.10) yields
α(n) ≤ α(n − 1) . e−
n+1
(ln( n+1
)−c)
4
2
.
(3.11)
The bounds (3.10) and (3.11) together show that (3.6) holds.
It remains to prove the boost in decay given in (3.9). If α(n) . (n + 1)−µ(n+1) and
3µ < 1/2, the self-consistency bound (3.2) gives
α(2n) . (n + 1)−3µ(n+1) + e−(n+1)(ln(n+1)−c)/2 . (n + 1)−3µ(n+1)
as long as 3µ < 1/2. Thus, as before, for even n one gets
3
3
n + 2 − 32 µ(n+2)
α(n) .
. (n + 2)−( 2 −ε)(n+2) ≤ (n + 1)−( 2 −ε)(n+1)
2
for any ε > 0. For odd n the monotonicity of α and (3.12) give
3
α(n) ≤ α(n − 1) . (n + 1)−( 2 −ε)(n+1)
(3.12)
(3.13)
for any ε > 0. The bounds (3.12) and (3.13) together show
3
α(n) . (n + 1)−( 2 −ε)(n+1)
(3.14)
for all n ∈ N0 and all ε > 0. Choosing ε = 1/4 yields (3.8).
Given Lemma 3.4, in order to prove the super–exponential decay of α given in
Theorem 3.2, it is enough to show that α(n) . (n + 1)−µ0 (n+1) for some arbitrarily
small µ0 > 0. This is the content of the next proposition
14
D. HUNDERTMARK AND Y.-R. LEE
Proposition 3.5. Assume that α is a non-negative decreasing function which obeys
the self-consistency bound (3.2) and goes to zero at infinity. Then there exists µ0 > 0
such that
α(n) . (n + 1)−µ0 (n+1)
For the proof of Proposition 3.5 we need some more notation. Given n ∈ N0 let
(n + 2)n+2 if n is even
F (n) :=
(3.15)
(n + 1)n+1 if n is odd
and, for ε ≥ 0, its regularized version
Fε (n) :=
Finally, for µ ≥ 0 let
F (n)
1
=
.
−1
1 + εF (n)
F (n) + ε
(3.16)
Fµ,ε (n) := Fε (n)µ = (F (n)−1 + ε)−µ .
(3.17)
kαkµ,ε,b := sup Fµ,ε (n)|α(n)|.
(3.18)
Furthermore let
n≥b
Of course, the super-exponential decay given in Proposition 3.5 is equivalent to showing
for some µ0 > 0 and some b ∈ N0 .
kαkµ0 ,0,b < ∞
Since kαkµ,0,b = sup0<ε≤1 kαkµ,ε,b , see Lemma 3.6.vi below, it is enough to find an
ε-independent bound on kαkµ,ε,b , which is where the second self-consistency bound
of Proposition 3.1 enters. First we gather some basic properties of Fµ,ε needed in the
proof of Proposition 3.5.
Lemma 3.6. (i) For any ε ≥ 0 the function n 7→ Fε (n) is increasing in n and, for
fixed n, decreasing in ε ≥ 0. Moreover, Fε (n) ≤ ε−1 for all n.
(ii) For any µ, ε ≥ 0 the function n 7→ Fµ,ε (n) is increasing and bounded by ε−µ .
Moreover, Fµ,ε (n) is decreasing in ε ≥ 0 and depends continuously on the parameters
µ and ε for fixed n ∈ N0 .
(iii) For any 0 ≤ µ ≤ δ/3, the function N0 ∋ n → Fµ,0 (2n)(n+1)−δ(n+1) is decreasing.
(iv) The bound
Fµ,ε (2n) ≤ 4Fµ,ε (n)3
(3.19)
holds for all 0 ≤ µ, ε ≤ 1 and n ∈ N0 .
(v) For fixed b ∈ N0 and an arbitrary bounded function α the map (µ, ε) 7→ kαkµ,ε,b
is continuous on [0, 1] × (0, 1].
(vi) For fixed 0 < µ, b ∈ N0 , and an arbitrary bounded function α,
kαkµ,0,b = lim kαkµ,ε,b = sup kαkµ,ε,b .
ε→0
0<ε≤1
Remark 3.7. The last part of the lemma shows that for fixed 0 < µ ≤ 1 and b ∈ N0
the map ε 7→ kαkµ,ε,b is continuous on [0, 1]. Here we interpret continuity in a generalized sense: One certainly has kαkµ,ε,b ≤ kαk∞ /ε < ∞ for all ε > 0. If kαkµ,0,b < ∞,
then limε→0 kαkµ,ε,b = kαkµ,0,b , and if kαkµ,0,b = ∞, then limε→0 kαkµ,ε,b = ∞.
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
15
We postpone the proof of Lemma 3.6 after the proof of Proposition 3.5.
Proof of Proposition 3.5. We assume that α decays monotonically to zero and obeys
the second self-consistency bound given in Proposition 3.1. That is, for fixed 0 < δ <
1/2 there exists a constant C0 such that
α(2n) ≤ C0 α(n)3 + C0 (n + 1)−δ(n+1) .
(3.20)
Multiply this by χ[b,∞), the characteristic function of the set [b, ∞), using
χ[b,∞)(n) = χ[b,∞)(2n) − χ[b,2b) (2n),
(3.21)
putting αb = χ[b,∞)α, and rearranging terms yields
αb (2n) ≤ C0 αb (n)3 + C0 (n + 1)−δ(n+1) χ[b,∞) (n) + χ[b,2b) (2n)α(2n).
(3.22)
Now let µ ≤ δ/3. Multiplying (3.22) by Fµ,ε (2n), using the bound (3.19) from Lemma
3.6 on the first term on the right hand side of (3.22) and Fµ,ε (2n) ≤ Fµ,0 (2n) from
Lemma 3.6.ii on the other two, assuming b ≥ 0, we get
3
Fµ,0 (2n)
χ[b,∞) (n)
Fµ,ε (2n)αb (2n) ≤ C1 Fµ,ε (n)αb (n) + C0
(n + 1)δ(n+1)
+ Fµ,0 (2n)χ[b,2b) (2n)α(2n)
3
(3.23)
Fµ,0 (2b)
≤ C1 Fµ,ε (n)αb (n) + C0
+
F
(2b)α(b)
µ,0
(b + 1)δ(b+1)
C0
3
≤ C1 kαkµ,ε,b + Fµ,0 (2b)
+ α(b) .
(b + 1)δ(b+1)
For the second inequality we used that α and, by Lemma 3.6.iii, Fµ,0 (2n)(n+1)−δ(n+1)
are decreasing and that Fµ,0 (2n) is increasing in n. In the third inequality, we used
the obvious bound
Fµ,ε (n)αb (n) ≤ kαkµ,ε,b
for all n ∈ N0 .
by the definition (3.18) for kαkµ,ε,b .
The punchline is that, since Fµ,ε (2n) = Fµ,ε (2n + 1) by construction of Fµ,ε , the
monotonicity of α gives
sup Fµ,ε (n)α(n) = sup Fµ,ε (2n)αb (2n)
n≥b
(3.24)
n∈N0
whenever b is an even natural number. So, since (3.23) holds for any n ∈ N0 , taking
the supremum over n in (3.23) yields
3
C
0
kαkµ,ε,b ≤ C1 kαµ,ε,b + Fµ,0 (2b)
+ α(b)
(3.25)
(b + 1)δ(b+1)
which is a closed inequality for kαkµ,ε,b and holds as long as b is even and 0 < µ ≤ δ/3
and 0 < ε ≤ 1 or µ = 0 and 0 ≤ ε ≤ 1.
Equivalently, with G(ν) := ν − C1 ν 3 defined for ν ≥ 0, we arrived at the a-priori
bound
G(kαkµ,ε,b ) ≤ Fµ,0 (2b) C0 (b + 1)−δ(b+1) + α(b) .
(3.26)
16
D. HUNDERTMARK AND Y.-R. LEE
A simple exercise shows that the maximum
of G is attained at νmax = (3C1 )−1/2 and is
√
given by Gmax = G(νmax ) = 2/(3 3C1 ). Let 0 < G0 < Gmax and 0 < ν0 < νmax < ν1
with G(ν0 ) = G(ν1 ) = G0 . Since G(ν) < ν for ν > 0, we have G(ν0 ) < ν0 . Moreover,
G−1 ((−∞, G0 ]) = [0, ν0 ] ∪ [ν1 , ∞).
(3.27)
This situation is visualized in Figure 1. Now we finish the proof of the decay estimate:
Gmax
G0
0
ν0
νmax
ν1
[0, ν0 ] ∪ [ν1 , ∞) = G−1 (−∞, G0 ]
Figure 1. Graph of G(ν) = ν − C1 ν 3 and the trapping region G−1 (−∞, G0 ] .
we need to show that kαkµ,0,b is finite for some µ > 0 and b ∈ N0 . Note that by Lemma
3.6.v the map (µ, ε) 7→ kαkµ,ε,b is continuous in (µ, ε) ∈ [0, 1] × (0, 1], and, by Lemma
3.6.vi, for fixed 0 < µ ≤ 1 kαkµ,0,b = limε→0 kαkµ,ε,b . So it will be enough to find, for
some µ > 0 and b ∈ N0 , a uniform in 0 < ε ≤ 1 estimate for kαkµ,ε,b . This is where
the bound (3.26) will enter.
Step 1: Choose an even b such that C0 (b + 1)−δ(b+1) + α(b) < G0 < Gmax . This
is possible since α goes to zero at infinity. Since α is monotone decreasing, it also
guarantees kαk0,1,b = supn∈N0 αb (n) = α(b) < G0 ≤ ν0 .
Step 2: For the b fixed in Step 1, let 0 < µ0 ≤ δ/3 such that Fµ0 ,0 (2b)(C0 (b +
1)−δ(b+1) + α(b)) ≤ G0 < Gmax and kαkµ0 ,1,b < ν0 . This is possible since F0,0 (2b) = 1
and Fµ,0 (2b) and kαkµ,1,b are continuous in 0 ≤ µ ≤ δ/3.
Putting things together, (3.26) gives
G(kαkµ0 ,ε,b ) ≤ G0
for all 0 < ε ≤ 1.
(3.28)
Since G is continuous and kαkµ0 ,ε,b depends continuously on ε > 0, the bound (3.28)
shows that kαkµ0 ,ε,b is trapped in the same connected component of G−1 ((−∞, G0 ])
as kαkµ0 ,1,b for all 0 < ε ≤ 1. Thus, using 0 ≤ kαkµ0 ,1,b < ν0 and (3.27), we must
have
kαkµ0 ,ε,b ≤ ν0 for all 0 < ε ≤ 1.
(3.29)
Together with kαkµ0 ,0,b = limε→0 kαkµ0 ,ε,b , the bound (3.29) shows kαkµ0 ,0,b ≤ ν0 <
∞ which proves the estimate
α(n) ≤ ν0 F (n)−µ0
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
17
for all large n. This finished our proof of Proposition 3.5.
It remains to prove the properties of Fµ,ε given in Lemma 3.6.
Proof of Lemma 3.6. The function F is clearly increasing in n and since s 7→ s/(1+εs)
is increasing in s ≥ 0 for fixed ε ≥ 0, we see that Fε and hence also Fµ,ε is increasing
in n. The other claims in part (i) and (ii) of Lemma 3.6 are obvious.
δ
− 1 ≥ 1/2, we have
(iii): With λ = 2µ
h
i2µ
Fµ,0 (2n)(n + 1)−δ(n+1) = (2(n + 1))2µ(n+1) (n + 1)−δ(n+1) = 2n+1 (n + 1)−λ(n+1)
.
Hence
"
#2µ
Fµ,0 (2(n + 1))(n + 2)−δ(n+2)
2
1 (n+1) −λ
=
1+
.
(n + 2)λ
n+1
Fµ,0 (2n)(n + 1)−δ(n+1)
1 n+1
is increasing, one has (1 + n+1
)
≥ 2 and
2µ
Fµ,0 (2(n + 1))(n + 2)−δ(n+2)
2
−λ
≤
2
≤ 2(1−2λ)2µ ≤ 1
(n + 2)λ
Fµ,0 (2n)(n + 1)−δ(n+1)
Since the sequence (1 +
1 n+1
n+1 )
so the function Fµ,0 (2n)(n + 1)−δ(n+1) is decreasing on N0 .
(iv): Put
Fε (2n)
(F (n)−1 + ε)3
.
f (n, ε) :=
=
Fε (n)3
F (2n)−1 + ε
We claim that
sup
f (n, ε) = 4,
which obviously yields Fε (2n)
n∈N0 , 0≤ε≤1
≤ 4Fε (n)3 and
Fµ,ε (2n) ≤ 4µ Fµ,ε (n)3 ≤ 4Fµ,ε (n)3
(3.30)
(3.31)
for all 0 ≤ µ ≤ 1.
So in order to prove (3.19) it is enough to show that (3.31) holds. The partial
derivative of f with respect to ε is given by
∂f
(F (n)−1 + ε)2 =
3F (2n)−1 − F (n)−1 + 2ε .
−1
2
∂ε
(F (2n) + ε)
In the case 3F (2n)−1 − F (n)−1 ≥ 0 one has ∂f
∂ε ≥ 0 for all ε ≥ 0 and the case
∂f
−1
−1
3F (2n) − F (n) < 0 one has ∂ε < 0 as long as 0 < ε < (F (n)−1 − 3F (2n)−1 )/2
−1 − 3F (2n)−1 )/2. Altogether, as a function of ε, f (n, ε)
and ∂f
∂ε > 0 for ε > (F (n)
is either increasing on [0, ∞) or it has a single minimum and no maximum in (0, ∞).
Hence, for fixed n ∈ N0 , its maximum in ε ∈ [0, 1] is attained at the boundary,
sup
n∈N0 , 0≤ε≤1
f (n, ε) = max( sup f (n, 0), sup f (n, 1)).
n∈N0
(3.32)
n∈N0
Since F1 (n) = F (n)/(1 + F (n)) = (1 + F (n)−1 )−1 and F (n) ≥ 4 for all n ∈ N0 ,
5 3
F1 (2n)
F (2n)
−1 3
f (n, 1) =
=
1
+
F
(n)
≤
<2
(3.33)
F1 (n)3
1 + F (2n)
4
18
D. HUNDERTMARK AND Y.-R. LEE
and using (n + 1)n+1 ≤ F (n) ≤ (n + 2)n+2 one sees
2(n+1)
n+1
2(n + 1)
4
F (2n)
f (n, 0) =
≤
=
≤4
F (n)3
n+1
(n + 1)3(n+1)
(3.34)
for all n ∈ N0 . Putting (3.32), (3.33), and (3.34) together and noticing f (1, 0) = 4
yields (3.31).
(v): Continuity of kαkµ,ε,b in (µ, ε) ∈ [0, 1] × (0, 1]: First note that the triangle
inequality implies
kαkµ ,ε ,b − kαkµ ,ε ,b ≤ sup (Fµ ,ε (n) − Fµ ,ε (n))α(n)
1
1
2
2
n≥b
1
1
2
2
−µ1
−µ2 ≤ kαk∞ sup F (n)−1 + ε1
− F (n)−1 + ε2
n∈N0
−µ1
−µ2 ≤ kαk∞ sup x + ε1
− x + ε2
(3.35)
0≤x≤1/4
since 4 ≤ F (n) for all n ∈ N0 . Let h(x, µ, ε) := (x + ε)−µ . For any 0 < ε′ < 1, h is
continuous on the compact set [0, 1/4]× [0, 1]× [ε′ , 1] and hence uniformly continuous.
Thus for any η > 0 there exists δ > such that for (xj , µj , εj ) ∈ [0, 1] × [0, 1] × [ε′ , 1],
j = 1, 2 with |x1 − x2 |, |µ1 − µ2 |, |ε1 − ε2 | < δ we have |h(x1 , µ1 , ε1 )− h(x2 , µ2 , ε2 )| < η.
In particular,
sup |h(x, µ1 , ε1 ) − h(x, µ2 , ε2 )| < η
0≤x≤1/4
which, together with (3.35), shows that (µ, ε) 7→ kαkµ,ε,b is uniformly continuous on
any compact subset of [0, 1] × (0, 1], hence continuous on [0, 1] × (0, 1].
(vi): Fix µ > 0. Recall that kαkµ,ε,b is decreasing in ε. Thus
lim kαkµ,ε,b = sup kαkµ,ε,b = sup sup Fµ,ε (n)|α(n)|
ε→0
0<ε≤1
0<ε≤1 n≥b
= sup sup Fµ,ε (n)|α(n)| = sup Fµ,0 (n)|α(n)| = kαkµ,0,b
n≥b 0<ε≤1
n≥b
which finishes the proof of Lemma 3.6.
4. Existence of Diffraction managed solitons for zero average
diffraction
Here we want to give a simple proof of existence of diffraction managed solitons, i.e.,
weak solutions of (1.11), via the direct method from the calculus of variations. This
hinges on the fact that the diffraction management equation is the Euler–Lagrange
equation for the functional
ϕ(f ) := Q(f, f, f, f )
(4.1)
2
on l (Z). The corresponding constraint maximization problem is given by
Pλ = sup ϕ(f ) : f ∈ l2 (Z), kf k22 = λ
(4.2)
where λ > 0. Up to some minor technical details it is clear that any maximizer f of
the variational problem (4.2), that is, any f ∈ l2 (Z) with kf k22 = λ and
Q(f, f, f, f ) = Pλ
(4.3)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
19
is by the Lagrange multiplier theorem a weak solution of the diffraction management
equation (1.11).
The usual way to show existence of a maximizer is to identify it as the limit of a suitable maximizing sequence, i.e., a sequence (fn )n with kfn k22 = λ and limn→∞ ϕ(fn ) =
Pλ . Such a sequence always exists, the problem, due to the translation invariance of
the time evolution Tt = eiD(t)∆ , is that the functional ϕ is invariant under translation;
if fn is a maximizing sequence for (4.2) and ξn any sequence in Z, then the shifted
sequence
fn,ξn (x) := fn (x − ξn )
is also a maximizing sequence. That is, the problem (4.2) is invariant under shifts,
yielding a loss of compactness since maximizing sequences can very easily converge
weakly to zero. The usual way to overcome this is the use of Lions’ concentration
compactness method [24] which, for positive average dispersion has been used in [27,
30]. For vanishing average diffraction, and under much more restrictive conditions on
the diffraction profile than (1.7), the existence of a maximizer of (4.2) has been shown
in [34] with the help of Ekeland’s variational principle, [12, 13], see also [17]. We will
give a different approach, which avoids the use of Lions’ concentration compactness
method or Ekeland’s variational principle by using the translation invariance of the
problem to show that for any maximizing sequence fn there exists a sequence of
shifts ξn such that the shifted sequence fn,ξn is tight in the sense of (4.4) below.
Then, since fn,ξn is bounded in l2 (Z), it has a weakly convergent and hence, by the
compactness Lemma 4.1 below, also a strongly convergent subsequence. The limit of
this subsequence is then a natural candidate for the maximizer of (4.2), see Theorem
4.7.
In the following, we will always assume that the diffraction profile obeys the bound
(1.7). Our main tools are the multi-linear bound from Corollary 2.9 and the following
simple compactness result.
Lemma 4.1. Let 1 ≤ p < ∞. A sequence (fn )n∈N ⊂ lp (Zd ) is strongly converging to
f in lp (Zd ) if and only if it is weakly convergent to f and the sequence is tight, i.e.,
X
|fn (x)|p = 0.
(4.4)
lim lim sup
L→∞ n→∞
|x|>L
Proof. For L ≥ 1, let KL be the operator of multiplication with the characteristic
function of [−L, L]d ∩ Zd , that is,
f (x) for |x| ≤ L
KL f (x) =
0
for |x| > L
and K L := 1 − KL . Note that both KL is an operator with finite dimensional range.
In particular, for any 1 ≤ L < ∞, KL : lp (Zd ) → lp (Zd ) is a compact operator.
Furthermore, both KL and K L are bounded operators with operator norm one since
p
kf kpp = kKL f kpp + kK L f kpp ≥ max kKL f kp , kK L f kp
20
D. HUNDERTMARK AND Y.-R. LEE
for all f ∈ lp (Zd ) and KL , respectively K L , acts as the identity on its range. Moreover,
the tightness-condition (4.4) is equivalent to
lim lim sup kK L fn kp = 0.
L→∞ n→∞
(4.5)
Now assume that fn converges strongly to f in lp (Zd ). Then it clearly converges also
weakly to f and
kK L fn kp ≤ kK L f kp + kK L (fn − f )kp ≤ kK L f kp + kfn − f kp .
Thus, by strong convergence of fn to f ,
lim sup kK L fn kp ≤ kK L f kp .
n→∞
Since f ∈
tight.
lp (Zd )
and p < ∞, we have limL→∞ kK L f kp = 0, so the sequence fn is
For the converse assume that fn converges to f weakly in lp (Zd ) and is tight, that is,
(4.5) holds. Then
kf − fn kp ≤ kKL (f − fn )kp + kK L (f − fn )kp ≤ kKL (f − fn )kp + kK L f kp + kK L fn kp .
Since KL is a compact operator on Lp , it maps weakly convergent sequences into
strongly converging sequences. Hence
lim sup kf − fn kp ≤ kK L f kp + lim sup kK L fn kp
n→∞
Now using f ∈
to get
lp (Zd )
n→∞
and the condition (4.5), take the limit L → ∞ in this inequality
lim sup kf − fn kp ≤ 0.
n→∞
Thus fn converges to f strongly in lp (Zd ).
Remark 4.2. The proof above breaks down for p = ∞, since for an arbitrary f ∈
l∞ (Zd ), one does not, in general, have
lim kK L f k∞ = 0.
L→∞
However, for the closed subspace l0∞ (Zd ) ⊂ l∞ (Z d ) consisting of bounded sequences
indexed by Zd vanishing at infinity, limx→∞ f (x) = 0 for any f ∈ l∞ (Zd ), the
above proof immediately generalizes and yields the analogous compactness statement:
(fn )n∈N converges strongly in l0∞ (Zd ) if and only if it converges weakly and
lim lim sup sup |fn (x)| = 0.
L→∞ n→∞ |x|>L
In the next Lemma we gather some simple bounds for the non-linear functional ϕ
and the associated constraint maximization problem (4.2).
Lemma 4.3. For any λ > 0 one has 0 < Pλ ≤ 1 and the scaling property Pλ = P1 λ2
holds. In particular, for any f ∈ l2 (Z) the bound
holds.
ϕ(f ) ≤ P1 kf k42
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
21
R1
Proof. Since ϕ(f ) = Q(f, f, f, f ) = 0 kTt f k44 dt we obviously have Pλ ≥ ϕ(f ) ≥ 0.
√
With Corollary 2.2, we see Pλ ≤ 1. By scaling, replacing f with fe = f / λ, one gets
Q(f, f, f, f ) = λ2 Q(fe, fe, fe, fe)
and taking the supremum over f with kf k22 = λ we see Pλ = P1 λ2 . If P1 = 0 then
R1
0 = Q(f, f, f, f ) = 0 kTt f k44 dt for all f ∈ l2 (Z) with kf k2 = 1. Thus, for almost
every t ∈ [0, 1] one would have Tt f (x) = 0 for all x ∈ Z. Hence f = 0 in contradiction
to kf k2 = 1. Thus P1 > 0.
The next lemma is key in our proof that maximizing sequences have strongly
convergent subsequences modulo translations. Its proof is inspired by the proof of
Lemma 2.3 in [20].
Lemma 4.4. Assume that the diffraction profile obeys the bound (1.7). Then there
is a constant C such that if f ∈ l2 (Z), ε2 < kf k22 and a, b ∈ Z, a ≤ b with
X
x<a
|f (x)|2 ≥
X
ε2
ε2
and
|f (x)|2 ≥ ,
2
2
(4.6)
x>b
then
ϕ(f ) ≤ P1 (kf k42 − ε4 /2) +
Ckf k42
1/2
(b − a + 1)1/2 − 1
(4.7)
Proof. The number of points in [a, b] ∩ Z is given by b − a + 1. Choose l ∈ N such
that
l ≤ (b − a + 1)1/2 < l + 1
(4.8)
and let I be a subinterval of {a, a + 1, . . . , b} consisting of l2 consecutive points. By
pigeonholing, there must be a subset I0 = {a′ , . . . , b′ } ⊂ I consisting of l consecutive
points with
X
x∈I0
|f (x)|2 ≤
kf k22
.
l
(4.9)
We split f into three parts, f−1 := f |(−∞,a′ ) , f1 := f |(b′ ,∞) , and f0 := f |I0 . Then
f = f−1 + f0 + f1 , kf k22 = kf−1 k22 + kf0 k22 + kf1 k22 , and (4.6) and (4.9) give
kf−1 k22 ≥
ε2
,
2
kf1 k22 ≥
ε2
,
2
kf k2
and kf0 k2 ≤ √ .
l
(4.10)
22
D. HUNDERTMARK AND Y.-R. LEE
Using the multi-linearity of Q,
X
Q(f, f, f, f ) =
Q(fj1 , fj2 , fj3 , fj4 )
jl ∈{−1,0,1}
l=1,...,4
X
=
jl ∈{−1,1}
l=1,...,4
Q(fj1 , fj2 , fj3 , fj4 ) +
X
jl ∈{−1,0,1}
l=1,...,4
some jl =0
Q(fj1 , fj2 , fj3 , fj4 )
= Q(f−1 , f−1 , f−1 , f−1 ) + Q(f1 , f1 , f1 , f1 )
X
X
+
Q(fj1 , fj2 , fj3 , fj4 ) +
jl ∈{−1,1}
∃k,l:jk =−1,jl =1
jl ∈{−1,0,1}
l=1,...,4
some jl =0
Q(fj1 , fj2 , fj3 , fj4 ).
(4.11)
Lemma 4.3 shows
Q(f−1 , f−1 , f−1 , f−1 ) + Q(f1 , f1 , f1 , f1 ) = ϕ(f−1 ) + ϕ(f1 ) ≤ P1 kf−1 k42 + kf1 k42
(4.12)
and, utilizing that the supports of f−1 and f1 have distance at least l + 1 ≥ 2, the
enhanced multi-linear bound (2.25) with the choice δ = 1/4 gives
X
1
kf k42
4
Q(fj , fj , fj , fj ) .
√ .
kf
k
≤
2
1
2
3
4
(4.13)
(l + 1)δ(l+1)
l
jl ∈{−1,1}
∃k,l:jk =−1,jl =1
Also,
X
jl ∈{−1,0,1}
l=1,...,4
some jl =0
Q(fj , fj , fj , fj ) ≤
1
2
3
4
X
4
Y
kf k4
kfjl k2 ≤ √ 2
l
j ∈{−1,0,1} l=1
l
l=1,...,4
some jl =0
(4.14)
by the a-priori-bound on Q given in Corollary 2.2 and (4.10) since each term in the
above sum contains at least one factor kf0 k2 . Thus (4.11) together with (4.12), (4.13),
and (4.14) gives
C
ϕ(f ) = Q(f, f, f, f ) ≤ P1 kf−1 k42 + kf1 k42 + √ kf k42
(4.15)
l
for some constant C. Since kf−1 k22 , kf1 k22 ≥ ε2 /2 and kf−1 k22 + kf1 k22 ≤ kf k22 , we have
kf−1 k42 + kf1 k42 ≤ kf k42 − 2kf−1 k22 kf1 k22 ≤ kf k42 − ε4 /2.
Together with (4.8) the bound (4.15) yields (4.7).
Lemma 4.5. Assume that the diffraction profile obeys the bound (1.7) and let (fn )n∈N
be a maximizing sequence for the constraint maximization
problem (4.2). Then there
√
exists a sequence (ξn )n∈N and, for each 0 < ε < λ, there exists Rε , which is independent of n, such that
X
lim sup
|fn (x)|2 ≤ ε2 .
(4.16)
n→∞
|x−ξn |>Rε
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
23
√
Proof. Given 0 < ε < λ as in the Lemma, we will show that there are corresponding
intervals In,ε = {cn,ε , . . . , dn,ε } which are nested,
In,ε ⊂ In,ε′
for all 0 < ε′ ≤ ε,
(4.17)
have bounded length,
sup(dn,ε − cn,ε ) < ∞,
(4.18)
n∈N
and contain most of the l2 norm of fn ,
X
lim sup
|fn (x)|2 ≤ ε2 .
n→∞
(4.19)
x6∈In,ε
Given (4.17),
√ (4.18), and (4.19), ξn and Rε can be constructed as follows: Fix some
0 < ε0 < λ. For each n ∈ N define ξn by simply choosing some point from the set
In,ε0 and define Rε by
Rε := sup(dn,ε − cn,ε )
n∈N
√
for ε ≤ ε0 and Rε =√Rε0 for ε0 < ε < λ. The bound (4.18) guarantees that Rε is
finite for all 0 < ε < λ. Since the intervals In,ε are nested in ε, the point ξn ∈ In,ε for
any 0 < ε ≤ ε0 . In particular, In,ε ⊂ {x : |x−ξn | ≤ Rε } for all 0 < ε ≤ ε0 by definition
of Rε . Moreover,√again since the sets In,ε are nested, In,ε ⊂ In,ε0 ⊂ {x : |x−ξn | ≤ Rε }
for all ε0 < ε < λ, too.
Putting everything together, (4.19) shows that with this choice of ξn and Rε the
bound (4.16) holds. Thus it is enough to prove (4.17), (4.18), and (4.19): Let
an,ε := min z ∈ Z :
bn,ε
X
x<z
|fn (x)|2 ≥
X
ε2 2
ε2 := max z ∈ Z :
|fn (x)| ≥
.
2
x>z
(4.20)
2
Both exist and are finite, since fn ∈ l2 (Z). Put cn,ε = min(an,ε , bn,ε ) − 1 and dn,ε =
max(an,ε , bn,ε ) + 1. With this choice (4.17) and (4.19) certainly hold and we only
have to check (4.18), which is where Lemma 4.4 enters.
Either an,ε > bn,ε , in which case dn,ε − cnε ≤ 2, or Lemma 4.4 applies and gives
the bound
ε2
Ckfn k42
P1 − Pλ − ϕ(fn ) ≤ (4.21)
1/2
2
1/2
(dn,ε − cn,ε − 1) − 1
where we rearranged (4.7) a bit and used the scaling P1 kfn k42 = P1 λ2 = Pλ . Since
fn is a maximizing sequence for (4.2) we have limn→∞ ϕ(fn ) = Pλ . Thus taking the
limit n → ∞ in (4.21) gives
P1
ε2
Cλ2
≤ lim inf 1/2
n→∞
2
(dn,ε − cn,ε − 1)1/2 − 1
24
D. HUNDERTMARK AND Y.-R. LEE
or
lim sup(dn,ε − cn,ε ) ≤
h 2Cλ2 2
P1 ε2
which proves (4.18) and hence the Lemma.
n→∞
i2
+ 1 + 1 < ∞.
A simple reformulation of Lemma 4.5 is
Corollary 4.6. Assume that the diffraction profile obeys the bound (1.7) and let
λ > 0. Then, given any maximizing sequence (fn )n of the variational problem (4.2),
there exists a sequence of translations ξn such that the translated sequence fn,ξn with
fn,ξn (x) := fn (x − ξn ) is tight, that is,
X
lim lim sup
|fn,ξn (x)|2 = 0.
(4.22)
L→∞ n→∞
|x|>L
Moreover, this shifted sequence is still a maximizing sequence for the variational problem (4.2).
Proof. By Lemma 4.5, the sequence fn,ξn is certainly tight. On the other hand it is
also a maximizing sequence for 4.2, since the time evolution Tt = eiD(t)∆ commutes
with translation, Tt fn,ξn (x) = Tt fn (x − ξn ) for all x, and hence
ϕ(fn,ξn ) = Q(fn,ξn , fn,ξn , fn,ξn , fn,ξn ) = Q(fn , fn , fn , fn ) = ϕ(fn )
for all n ∈ N.
Theorem 4.7 (= Theorem 1.3). Let λ > 0 and assume that the diffraction profile
obeys the bound (1.7). Then there is an f ∈ l2 (Z) with kf k22 = λ such that
ϕ(f ) = Pλ :=
sup
ϕ(g).
g:kgk22 =λ
This maximizer f is also a weak solution of the dispersion management equation
ωf = Q(f, f, f ).
where ω = Pλ /λ > 0 is the Lagrange-multiplier.
Proof. Let λ > 0 and (fn )n be a maximizing sequence for (4.2) with kfn k2 = λ.
By Corollary 4.6, we can, without loss of generality, assume that this maximizing
sequence is already tight in the sense of equation (4.22).
Since the unit ball in l2 (Z) is weakly compact, there is a subsequence (fnj )j∈N of
(fn )n∈N which converges weakly to some f ∈ l2 (Z). From Lemma 4.1 we know that
fnj converges even strongly to f in l2 (Z). Thus kf k22 = λ and hence f is a good
candidate for the maximizer. To conclude that f is a maximizer for the variational
problem, we need to show that ϕ(f ) = Pλ . Since kf k22 = λ one certainly has
ϕ(f ) ≤ Pλ = lim ϕ(fn ),
n→∞
so one only needs upper semi-continuity of ϕ at f , i.e.,
lim sup ϕ(fnj ) ≤ ϕ(f ).
n→∞
(4.23)
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
25
By Lemma 4.8 below, the map f 7→ ϕ(f ) is even continuous on l2 (Z), in particular, (4.23) is true which finished the proof that the variational problem (4.2) has a
maximizer.
The proof that the above maximizer is a weak solution of the associated Euler–
Lagrange equation (1.11) is standard in the calculous of variations, we sketch it for
the convenience of the reader: Lemma 4.9 below shows that the derivative of the
functional ϕ at any f ∈ l2 (Z) is given by the linear map Dϕ(f )[h] = 4ReQ(h, f, f, f ).
Similarly, one can check that the derivative of ψ(f ) = kf k22 = hf, f i is given by
Dψ(f )[h] = 2Rehh, f i. Note that although, in our convention for the inner product,
the map h 7→ hh, f i is anti-linear, the map h 7→ Rehh, f i is linear. Similarly, one
easily checks that for fixed f the map h 7→ ReQ(h, f, f, f ) is linear.
Now let f be any maximizer of the constraint variational problem (4.2) and h ∈
l2 (Z) arbitrary. Define, for any (s, t) ∈ R2 ,
F (s, t) := ϕ(f + sf + th),
G(s, t) := ψ(f + sf + th).
Note that
Dϕ(f + sf + th)[f ]
∇F (s, t) =
Dϕ(f + sf + th)[h]
ReQ(f, f + sf + th, f + sf + th, f + sf + th)
=4
ReQ(h, f + sf + th, f + sf + th, f + sf + th)
and
∇G(s, t) =
Since hf, f i = λ 6= 0,
Dψ(f + sf + th)[f ]
Dψ(f + sf + th)[h]
∇G(0, 0) = 2
=2
hf, f i
Rehh, f i
Rehf, f + sf + thi
Rehh, f + sf + thi
.
is not the zero vector in R2 and since ∇G(s, t) depends multi-linearly, in particular
continuously, on (s, t), the implicit function theorem [36] shows that there exists an
open interval I ⊂ R containing 0 and a differentiable function φ on I with φ(0) = 0
such that
λ = kf k22 = G(0, 0) = G(φ(t), t)
for all t ∈ I. Consider the function I ∋ t 7→ F (φ(t), t). Since f is a maximizer for
the constraint variational problem (4.2), F (φ(t), t) has a local maximum at t = 0.
Hence, using the chain rule,
′
dF (φ(t), t) φ (0)
= ∇F (0, 0) ·
= 4Q(f, f, f, f )φ′ (0) + 4ReQ(h, f, f, f )
0=
1
dt
t=0
Since λ = G(φ(t), t), the chain rule also yields
′
dG(φ(t), t) φ (0)
0=
=
∇G(0,
0)
·
= 2hf, f iφ′ (0) + 2Rehh, f i.
1
dt
t=0
26
D. HUNDERTMARK AND Y.-R. LEE
Solving this for φ′ (0) and plugging it back into the expression for the derivative of F ,
we see that
Q(f, f, f, f )
Rehh, f i = ReQ(h, f, f, f ).
hf, f i
In other words, with ω := Q(f, f, f, f )/hf, f i = Pλ /λ > 0 and f any maximizer of
(4.2), we have
Re(ωhh, f i) = ReQ(h, f, f, f )
(4.24)
2
for any h ∈ l (Z). Replacing h by ih in (4.24), one gets
for all h ∈
l2 (Z).
for any h ∈
(1.11).
l2 (Z),
Im(ωhh, f i) = ImQ(h, f, f, f )
(4.25)
(4.24) and (4.25) together show
ωhh, f i = Q(h, f, f, f )
that is, f is a weak solution of the diffraction management equation
In the proof of Theorem 4.7, we needed the following two Lemmata.
Lemma 4.8. The map f 7→ ϕ(f ) = Q(f, f, f, f ) is locally Lipshitz continuous on
l2 (Z).
Proof. Given f, g, one has
Q(f, f, f, f ) − Q(g, g, g, g) =
=
Z
1
0
Z
kTt f k44 − kTt gk44 dt
3
1X
0 j=0
kTt f k3−j
4
kTt f k4 −
kTt gk4 kTt gkj4
(4.26)
dt .
The above together with the triangle inequality, the bound khk4 ≤ khk2 for all
h ∈ l2 (Z), see Lemma 2.1, and the unicity of Tt on l2 (Z) gives
Z 1X
3
Q(f, f, f, f ) − Q(g, g, g, g) ≤
kTt f k2 − kTt gk2 kTt gkj dt
kTt f k3−j
2
2
0 j=0
≤
Z
≤
3
X
3
1X
0 j=0
j=0
j
kTt f k3−j
2 kTt (f − g)k2 kTt gk2 dt
(4.27)
j
kf k3−j
2 kf − gk2 kgk2
≤ 4 max(1, kf k32 , kgk32 )kf − gk2 .
We need one more technical result, about the differentiability of the non-linear
functional ϕ.
Lemma 4.9. The functional ϕ is continuously differentiable with derivative Dϕ(f )[h] =
4ReQ(h, f, f, f )
SUPER-EXPONENTIAL DECAY OF DIFFRACTION MANAGED SOLITONS
27
Proof. Using the multi-linearity of Q, one can check that for any f, h ∈ l2 (Z)
ϕ(f + h) = ϕ(f ) + Q(f, f, f, h) + Q(f, f, h, f ) + Q(f, h, f, f ) + Q(h, f, f, f ) + O(khk22 )
= ϕ(f ) + 4ReQ(h, f, f, f ) + O(khk22 )
(4.28)
where in the term O(khk22 ) we gathered expressions of the form Q(h, f, f, h), and
Q(h, f, f, f + h), and similar, which by Corollary 2.2 are bounded by Ckhk22 . This
shows that ϕ is differentiable with derivative Dϕ(f )[h] = 4ReQ(h, f, f, f ). Moreover,
Dϕ(f )[h] − Dϕ(g)[h] = 4Re Q(h, f, f, f ) − Q(h, g, g, g)
= 4Re Q(h, f − g, f, f ) + Q(h, g, f − g, f ) + Q(h, g, g, f − g) .
Hence using the bound from Corollary 2.2 again, we see
sup Dϕ(f )[h] − Dϕ(g)[h] ≤ 4(kf k22 + kf k2 kgk2 + kgk22 )kf − gk2
(4.29)
khk2 ≤1
which shows that the derivative Dϕ is even locally Lipshitz continuous.
Acknowledgment: It is a pleasure to thank Vadim Zharnitsky for introducing us
to the dispersion management technique. Young-Ran Lee thanks the Departments
of Mathematics of the University of Alabama at Birmingham and the University
of Illinois at Urbana-Champaign for their warm hospitality during her visit when
most of this work was done. Young-Ran Lee was partially supported by the Korean
Science and Engineering Foundation(KOSEF) grant funded by the Korean government(MOST) (No.R01-2007-000-11307-0).
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Department of Mathematics, Altgeld Hall, and Institute for Condensed Matter
Theory at the University of Illinois at Urbana–Champaign, 1409 W. Green Street,
Urbana, IL 61801.
E-mail address: dirk@math.uiuc.edu
Department of Mathematical Sciences KAIST (Korea Advanced Institute of Science
and Technology), 335 Gwahangno, Yuseong-gu, Daejeon, 305-701, Republic of Korea.
E-mail address: youngranlee@kaist.ac.kr